Theorem numclwwlkovf2 27217

Description: Value of operation 𝐹 for argument 2. (Contributed by Alexander van der Vekens, 19-Sep-2018.) (Revised by AV, 28-May-2021.)

Hypotheses

Ref	Expression
numclwwlkovf.f	⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
numclwwlkffin.v	⊢ 𝑉 = (Vtx‘𝐺)
numclwwlkovfel2.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
numclwwlkovf2	⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → (𝑋𝐹2) = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋)})

Distinct variable groups:   𝑛,𝐺,𝑣,𝑤   𝑛,𝑉,𝑣   𝑛,𝑋,𝑣,𝑤   𝑤,𝑉

Allowed substitution hints:   𝐸(𝑤,𝑣,𝑛)   𝐹(𝑤,𝑣,𝑛)

Proof of Theorem numclwwlkovf2

Step	Hyp	Ref	Expression
1		simpr 477	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉)
2		2nn 11185	. . 3 ⊢ 2 ∈ ℕ
3		numclwwlkovf.f	. . . 4 ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
4	3	numclwwlkovf 27213	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 2 ∈ ℕ) → (𝑋𝐹2) = {𝑤 ∈ (2 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋})
5	1, 2, 4	sylancl 694	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → (𝑋𝐹2) = {𝑤 ∈ (2 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋})
6		clwwlksn2 26910	. . . . . 6 ⊢ (𝑤 ∈ (2 ClWWalksN 𝐺) ↔ ((#‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)))
7	6	anbi1i 731	. . . . 5 ⊢ ((𝑤 ∈ (2 ClWWalksN 𝐺) ∧ (𝑤‘0) = 𝑋) ↔ (((#‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋))
8	7	a1i 11	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → ((𝑤 ∈ (2 ClWWalksN 𝐺) ∧ (𝑤‘0) = 𝑋) ↔ (((#‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)))
9		anass 681	. . . . 5 ⊢ (((𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸)) ∧ (𝑤‘0) = 𝑋) ↔ (𝑤 ∈ Word 𝑉 ∧ (((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸) ∧ (𝑤‘0) = 𝑋)))
10		df-3an 1039	. . . . . . . 8 ⊢ (((#‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ↔ (((#‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺)) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)))
11		ancom 466	. . . . . . . . . 10 ⊢ (((#‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺)) ↔ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 2))
12		numclwwlkffin.v	. . . . . . . . . . . . . 14 ⊢ 𝑉 = (Vtx‘𝐺)
13	12	eqcomi 2631	. . . . . . . . . . . . 13 ⊢ (Vtx‘𝐺) = 𝑉
14	13	wrdeqi 13328	. . . . . . . . . . . 12 ⊢ Word (Vtx‘𝐺) = Word 𝑉
15	14	eleq2i 2693	. . . . . . . . . . 11 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) ↔ 𝑤 ∈ Word 𝑉)
16	15	anbi1i 731	. . . . . . . . . 10 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 2) ↔ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = 2))
17	11, 16	bitri 264	. . . . . . . . 9 ⊢ (((#‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺)) ↔ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = 2))
18		numclwwlkovfel2.e	. . . . . . . . . . 11 ⊢ 𝐸 = (Edg‘𝐺)
19	18	eqcomi 2631	. . . . . . . . . 10 ⊢ (Edg‘𝐺) = 𝐸
20	19	eleq2i 2693	. . . . . . . . 9 ⊢ ({(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺) ↔ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸)
21	17, 20	anbi12i 733	. . . . . . . 8 ⊢ ((((#‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺)) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ↔ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = 2) ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸))
22	10, 21	bitri 264	. . . . . . 7 ⊢ (((#‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ↔ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = 2) ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸))
23		anass 681	. . . . . . 7 ⊢ (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = 2) ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸) ↔ (𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸)))
24	22, 23	bitri 264	. . . . . 6 ⊢ (((#‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ↔ (𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸)))
25	24	anbi1i 731	. . . . 5 ⊢ ((((#‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) ↔ ((𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸)) ∧ (𝑤‘0) = 𝑋))
26		df-3an 1039	. . . . . 6 ⊢ (((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋) ↔ (((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸) ∧ (𝑤‘0) = 𝑋))
27	26	anbi2i 730	. . . . 5 ⊢ ((𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋)) ↔ (𝑤 ∈ Word 𝑉 ∧ (((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸) ∧ (𝑤‘0) = 𝑋)))
28	9, 25, 27	3bitr4i 292	. . . 4 ⊢ ((((#‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) ↔ (𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋)))
29	8, 28	syl6bb 276	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → ((𝑤 ∈ (2 ClWWalksN 𝐺) ∧ (𝑤‘0) = 𝑋) ↔ (𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋))))
30	29	rabbidva2 3186	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → {𝑤 ∈ (2 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋)})
31	5, 30	eqtrd 2656	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → (𝑋𝐹2) = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋)})

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  {crab 2916  {cpr 4179  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  0cc0 9936  1c1 9937  ℕcn 11020  2c2 11070  #chash 13117  Word cword 13291  Vtxcvtx 25874  Edgcedg 25939   USGraph cusgr 26044   ClWWalksN cclwwlksn 26876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-clwwlks 26877  df-clwwlksn 26878

This theorem is referenced by:  numclwwlkovf2num  27218